Moving One-Component Regularity Criteria for the 3D Incompressible MHD Equations

Abstract

We establish a scaling-critical continuation criterion for the three-dimensional incompressible magnetohydrodynamic equations in ${\mathbb{R}}^3$ with arbitrary positive viscosity and magnetic diffusivity. Let $\beta (t)$ be a unit vector that is piecewise $H^1$ in time and has only finitely many jumps. For $3\le p<\infty$, set ${\gamma }_p=2p/(2p-3)$. We prove that an $H^1$ strong solution can be continued beyond $T_{\ast }$ if \[ \int_0^{T_*}\left( \|u(t)\cdotβ(t)\|_{\dot H^{3/2}}^2+ \|b(t)\cdotβ(t)\|_{\dot H^{3/2}}^2+ \|β(t)\cdot\operatorname{curl}b(t)\|_{L^p}^{γ_p} \right)\,dt<\infty . \] Thus the observed component may vary in time, and the magnetic assumption is reduced to one moving magnetic component together with one Serrin-type current-density component. The proof is based on a moving-frame formulation of the vorticity-current system, an anisotropic product estimate adapted to the moving frame, and a horizontal Hodge decomposition that controls the current-Jacobian residual.